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Claim. Alper Ferudun, arXiv:2606.28041v1, submitted 26 June 2026 (the claim's date), asserts in its Theorem 1.1 that for every integer $1\le n\le40$ the largest number of edge deletions a triangle-free graph on $5n$ vertices needs to become bipartite is exactly $n^2$. The upper bound answers the question of Problem 23 yes for those $n$, and the balanced blow-up of $C_5$ (five stable sets of $n$ vertices, complete bipartite graphs between cyclically consecutive sets) needs $n^2$ deletions, so the value is exact. Ferudun also stated the values $a(5k)=k^2$ for $1\le k\le40$ in a comment on OEIS A389646 dated 29 June 2026, citing the preprint. The certificates and the full proof are not independently reviewed. The introduction also contains prose that does not agree with the theorem's range (it speaks of eleven multiples of five and of $n\ge12$), recorded as an unresolved textual inconsistency.

Covers. The question for graphs on $5n$ vertices with $1\le n\le40$. Nothing for $n>40$: the problem asks about every $n$, and neither a proof for all $n$ nor a counterexample follows. The claim combines the two edge-density tails of Balogh, Clemen and Lidický's Theorem 2(b),(c) with Ferudun's own certificate for the middle band: Ferudun's Theorem 4.1 cites their theorem, at most $N^2/25$ deletions for edge density at most $0.2486$ or at least $0.3197$ once $N$ is sufficiently large, and the paper's Corollary 4.2 carries it to each $N\le200$ by blowing up, through the identity $\beta(G[t])=t^2\beta(G)$ for the deletion number of the $t$-fold blow-up; an order-10 certificate then covers the densities between the two thresholds. Only their general bound of $N^2/23.5$ deletions, that is $(50/47)n^2$, is cited and not used.

Depends on. Theorem 2(b),(c) of Balogh, Clemen and Lidický, for the two edge-density tails; nothing else in this wiki.

Standing. Claimed: a preprint with no journal record and no independent review found in the search that the problem page records; the site's proof-claim tab carried no entry for this problem on 2026-10-06. The claim is partial, so the problem's standing is unchanged by it.